Follow the money. Selling taxi medallions is a huge source of revenue and graft.
That is true but the summary refers to Brooklyn and Queens, a.k.a. "outer boroughs" (anything that isn't Manhattan). The outer boroughs now have "Green Taxis" which do not bear medallions, and there are about 15,000 of them so far:
NYC also has "livery" cabs which can be summoned via phone, in contrast to "taxis" which are hailed on the street. Livery cabs don't bear medallions either.
The concerns about Lyft and Uber probably is more about the proper training and licensing of drivers, liability insurance coverage, adherance to laws (like non-discrimination in picking up passengers, and like fair labor practices). Not medallions.